How can I cut an edge at the mathematical middle?
Subdivide the edge

Subdivide the edge by clicking Subdivide on the Tools Panel (T key) or by searching the Subdivide operator in the Operator Search (space key)

  

The edge is cut in 2

Now the edge is cut in 2 by its middle - a new vertex is added.



How have more cuts?

If you want to have more vertices equally distributed, simply change the Number of Cuts in the Subdivide properties